Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Prove that there are infinitely many such $n$ such that Bob has a winning strategy. (For example, if $n=17,$ then Alice might take $6$ leaving $11;$ then Bob might take $1$ leaving $10;$ then Alice can take the remaining stones to win.)

Box A contains $3$ black and $4$ blue marbles. Box B has $7$ black and $1$ blue, whereas Box C has $2$ black, $3$ blue and $1$ green marble. I close my eyes and pick two marbles from $2$ different boxes. If it turns out that I get $1$ black and $1$ blue marble, what is the probability that the black marble is from box A and the blue one is from C?

In a small pond there are eleven lily pads in a row labeled $0$ through $10$. A frog is sitting on pad $1$. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad $0$ it will be eaten by a patiently waiting snake. If the frog reaches pad $10$ it will exit the pond, never to return. What is the probability that the frog will escape being eaten by the snake? $ \textbf {(A) } \frac{32}{79} \qquad \textbf {(B) } \frac{161}{384} \qquad \textbf {(C) } \frac{63}{146} \qquad \textbf {(D) } \frac{7}{16} \qquad \textbf {(E) } \frac{1}{2} $

Kelvin the Frog is going to roll three fair ten-sided dice with faces labelled $0, 1, \dots, 9$. First he rolls two dice, and finds the sum of the two rolls. Then he rolls the third die. What is the probability that the sum of the first two rolls equals the third roll?

An envelope contains eight bills: $ 2$ ones, $ 2$ fives, $ 2$ tens, and $ 2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $ \$ 20$ or more? $ \textbf{(A)}\ \frac {1}{4}\qquad \textbf{(B)}\ \frac {2}{7}\qquad \textbf{(C)}\ \frac {3}{7}\qquad \textbf{(D)}\ \frac {1}{2}\qquad \textbf{(E)}\ \frac {2}{3}$

[5] If four fair six-sided dice are rolled, what is the probability that the lowest number appearing on any die is exactly $3$?

Alice throws two standard dice, with $A$ being the number on her first die and $B$ being the number on her second die. She then draws the line $Ax + By = 2013$. Boris also throws two standard dice, with $C$ being the number on his first die and $D$ being the number on his second die. He then draws the line $Cx + Dy = 2014$. Compute the probability that these two lines are parallel.

Let $c$ be the probability that the cards are neither from the same suit or the same rank. Compute $\lfloor 1000c\rfloor$.

A coastal battery sights an enemy cruiser lying one kilometer off the coast and opens fire on it at the rate of one round per minute. After the first shot, the cruiser begins to move away at a speed of $ 60$ kilometers an hour. Let the probability of a hit be $ 0.75x^{ \minus{} 2}$, where $ x$ denotes the distance (in kilometers) between the cruiser and the coast ($ x\geq 1$), and suppose that the battery goes on firing till the cruiser either sinks or disappears. Further, let the probability of the cruiser sinking after $ n$ hits be $ 1 \minus{} \frac {1}{4^n}$ ($ n \equal{} 0,1,...$). Show that the probability of the cruiser escaping is $ \frac {2\sqrt {2}}{3\pi}$

First $a$ is chosen at random from the set $\{1,2,3,\ldots,99,100 \}$, and then $b$ is chosen at random from the same set. The probability that the integer $3^a+7^b$ has units digit $8$ is $\text{(A)} \ \frac1{16} \qquad \text{(B)} \ \frac18 \qquad \text{(C)} \ \frac{3}{16}\qquad \text{(D)} \ \frac15 \qquad \text{(E)} \ \frac14$

Let $ABC$ be a triangle. Choose points $A'$, $B'$ and $C'$ independently on side segments $BC$, $CA$ and $AB$ respectively with a uniform distribution. For a point $Z$ in the plane, let $p(Z)$ denote the probability that $Z$ is contained in the triangle enclosed by lines $AA'$, $BB'$ and $CC'$. For which interior point $Z$ in triangle $ABC$ is $p(Z)$ maximised?

Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpositions? $(\textbf{A})\: 1.6\qquad(\textbf{B}) \: 1.8\qquad(\textbf{C}) \: 2.0\qquad(\textbf{D}) \: 2.2\qquad(\textbf{E}) \: 2.4$

Let $a, b, c, d$ be a permutation of the numbers $1, 9, 8,4$ and let $n = (10a + b)^{10c+d}$. Find the probability that $1984!$ is divisible by $n.$

An unfair coin has the property that when flipped four times, it has the same nonzero probability of turning up $2$ heads and $2$ tails (in any order) as $3$ heads and $1$ tail (in any order). What is the probability of getting a head in any one flip?

Among $100$ points in the plane, no three collinear, exactly $4026$ pairs are connected by line segments. Each point is then randomly assigned an integer from $1$ to $100$ inclusive, each equally likely, such that no integer appears more than once. Find the expected value of the number of segments which join two points whose labels differ by at least $50$. [i]Proposed by Evan Chen[/i]

A black bishop and a white king are placed randomly on a $2000 \times 2000$ chessboard (in distinct squares). Let $p$ be the probability that the bishop attacks the king (that is, the bishop and king lie on some common diagonal of the board). Then $p$ can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m$. [i]Proposed by Ahaan Rungta[/i]

A robot on the number line starts at $1$. During the first minute, the robot writes down the number $1$. Each minute thereafter, it moves by one, either left or right, with equal probability. It then multiplies the last number it wrote by $n/t$, where $n$ is the number it just moved to, and $t$ is the number of minutes elapsed. It then writes this number down. For example, if the robot moves right during the second minute, it would write down $2/2=1$. Find the expected sum of all numbers it writes down, given that it is finite. [i]Proposed by Ethan Tan[/i]

Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble?

Let $K{}$ be an equilateral triangle of unit area, and choose $n{}$ independent random points uniformly from $K{}$. Let $K_n$ be the intersection of all translations of $K{}$ that contain all the selected points. Determine the expected value of the area of $K_n.$

An equilateral triangle is tiled with $n^2$ smaller congruent equilateral triangles such that there are $n$ smaller triangles along each of the sides of the original triangle. For each of the small equilateral triangles, we randomly choose a vertex $V$ of the triangle and draw an arc with that vertex as center connecting the midpoints of the two sides of the small triangle with $V$ as an endpoint. Find, with proof, the expected value of the number of full circles formed, in terms of $n.$ [img]http://s3.amazonaws.com/classroom.artofproblemsolving.com/Images/Transcripts/497b4e1ef5043a84b433a5c52c4be3ae.png[/img]

A fair 100-sided die is rolled twice, giving the numbers $a$ and $b$ in that order. If the probability that $a^2-4b$ is a perfect square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $100m+n$. [i] Proposed by Justin Stevens [/i]

You are tossing an unbiased coin. The last $ 28 $ consecutive flips have all resulted in heads. Let $ x $ be the expected number of additional tosses you must make before you get $ 60 $ consecutive heads. Find the sum of all distinct prime factors in $ x $.

Let $m/n$, in lowest terms, be the probability that a randomly chosen positive divisor of $10^{99}$ is an integer multiple of $10^{88}$. Find $m + n$.

In a tennis ball factory there are $4$ machines $m_1 , m_2 , m_3 , m_4$, which produce, respectively, $10\%$, $20\%$, $30\%$ and $40\%$ of the balls that come out of the factory. The machine $m_1$ introduces defects in $1\%$ of the balls it manufactures, the machine $m_2$ in $2\%$, $m_3$ in $4\%$ and $m_4$ in $15\%$. Of the balls manufactured In one day, one is chosen at random and it turns out to be defective. What is the probability that Has this ball been made by the machine $ m_3$ ?

During the regular season, Washington Redskins achieve a record of $10$ wins and $6$ losses. Compute the probability that their wins came in three streaks of consecutive wins, assuming that all possible arrangements of wins and losses are equally likely. (For example, the record $LLWWWWWLWWLWWWLL$ contains three winning streaks, while $WWWWWWWLLLLLLWWW$ has just two.)